NAG FL Interface
f07qrf (zsptrf)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{uplo}$ – Character(1) Input

On entry: specifies whether the upper or lower triangular part of $A$ is stored and how $A$ is to be factorized.

$\mathbf{uplo}=\text{'U'}$

The upper triangular part of $A$ is stored and $A$ is factorized as $PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $U$ is upper triangular.

$\mathbf{uplo}=\text{'L'}$

The lower triangular part of $A$ is stored and $A$ is factorized as $PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $L$ is lower triangular.

Constraint: $\mathbf{uplo}=\text{'U'}$ or $\text{'L'}$.

2: $\mathbf{n}$ – Integer Input

On entry: $n$, the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 0$.

3: $\mathbf{ap}\left(*\right)$ – Complex (Kind=nag_wp) array Input/Output

Note: the dimension of the array ap must be at least $\max (1,\mathbf{n}\times (\mathbf{n}+1)/2)$.

On entry: the $n\times n$ symmetric matrix $A$, packed by columns.

More precisely,

• if $\mathbf{uplo}=\text{'U'}$, the upper triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{ap}\left(i+j(j-1)/2\right)$ for $i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the lower triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{ap}\left(i+(2n-j)(j-1)/2\right)$ for $i\ge j$.

On exit: $A$ is overwritten by details of the block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ as specified by uplo.

4: $\mathbf{ipiv}\left(\mathbf{n}\right)$ – Integer array Output

On exit: details of the interchanges and the block structure of $D$. More precisely,

• if $\mathbf{ipiv}\left(i\right)=k>0$, $d_{ii}$ is a $1\times 1$ pivot block and the $i$th row and column of $A$ were interchanged with the $k$th row and column;
• if $\mathbf{uplo}=\text{'U'}$ and $\mathbf{ipiv}\left(i-1\right)=\mathbf{ipiv}\left(i\right)=-l<0$, $\left(\begin{array}{cc}{d}_{i-1,i-1}& {\overline{d}}_{i,i-1}\\ {\overline{d}}_{i,i-1}& d_{ii}\end{array}\right)$ is a $2\times 2$ pivot block and the $(i-1)$th row and column of $A$ were interchanged with the $l$th row and column;
• if $\mathbf{uplo}=\text{'L'}$ and $\mathbf{ipiv}\left(i\right)=\mathbf{ipiv}\left(i+1\right)=-m<0$, $\left(\begin{array}{cc}{d}_{ii}& d_{i+1,i}\\ d_{i+1,i}& d_{i+1,i+1}\end{array}\right)$ is a $2\times 2$ pivot block and the $(i+1)$th row and column of $A$ were interchanged with the $m$th row and column.

5: $\mathbf{info}$ – Integer Output

On exit: $\mathbf{info}=0$ unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$\mathbf{info}<0$

If $\mathbf{info}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$\mathbf{info}>0$

Element $⟨\mathit{\text{value}}⟩$ of the diagonal is exactly zero. The factorization has been completed, but the block diagonal matrix $D$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results